Final answer:
The equation for the cosine function with an amplitude of 2/3, a period of 1.8, a phase shift of -5.2, and a vertical shift of 3.9 is y(t) = (2/3) cos((2π / 1.8)(t + 5.2)) + 3.9.
Step-by-step explanation:
To write an equation of the cosine function with given conditions, we first recall the general form of a cosine function:
y(t) = A cos(B(t - C)) + D
Where:
A is the amplitude of the wave
B determines the period of the wave
C is the phase shift
D is the vertical shift
Given:
Amplitude A = 2/3
Period T = 1.8, which means B = 2π/T
Phase shift C = -5.2
Vertical shift D = 3.9
To find B, we use the fact that the period T is the reciprocal of the frequency f (so f = 1/T), and since B = 2πf, we have:
B = 2π * (1/1.8) = 2π / 1.8
Now, plugging in the given values into the cosine function, we get:
y(t) = (2/3) cos((2π / 1.8)(t + 5.2)) + 3.9
It is worth noting that to account for the phase shift to the left, we add the phase shift value to t inside the cosine function.